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In the first case, YES they are in series. In the second diagram, the resistors are in series so far as battery #1 is concerned, as you have drawn it. You have a faulty diagram, however, since you have shorted out battery #2. I assume instead that you meant the bottom of battery #2 to connect to the node at far left where both resistors meet. In that case, the circuit is neither purely series nor parallel, but more complicated. The voltages and currents can be found by applying Kirchoff's laws, if you have come across those.

So I guess the takeaway of this, is it the case that a capacitor and/or battery that is in between two resistors doesn't affect whether you consider them as being in series or parallel? For a capacitor, the current can't go through the air in between, so of course they don't have the same current, but are they still in series?

The terms 'series' and 'parallel' are not the be all and end all of circuit analysis. Don't get hung up on how to use them. The point of describing a pair of components as being 'in parallel' is when you are considering the way current splits between them. When you are considering two components 'in line' and how the voltage is shared between them, you use the term 'series'. In any particular circuit, it is often quite possible to treat two components as either series or parallel, depending on what you want to find out.
Avoid the 'classification bug'.